Abstract

A study is made of the effect of diffraction on imaging a square-wave object (in intensity) in incoherent light for slit and circular apertures suffering from spherical aberration. A modification of the usual Fourier series representation of the square wave is required to eliminate the unwanted Gibbs phenomena. The modulation of the image of the square wave is calculated by using transfer-function theory. Numerous examples are presented.

© 1963 Optical Society of America

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References

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  1. L. Foucault, Ann. l’Observ. Imp. Paris 5, 197 (1858).
  2. Rayleigh (J. W. Strutt), Phil. Mag. 42, 167 (1896).
  3. M. Françon, article in Handbuch der Physik, edited by S. Flügge (Springer-Verlag, Berlin, 1956), Vol. XXIV, pp. 171, 342; A. Marechal and M. Françon, Diffraction—Structure des images (Editions de la Revue d’Optique, Paris, 1960), pp. 38, 50, 169.
    [Crossref]
  4. R. Barakat and M. V. Morello, J. Opt. Soc. Am. 52, 992 (1962).
    [Crossref]
  5. H. S. Carslaw, Introduction to the Theory of Fourier’s Series and Integrals (Dover Publications, Inc., New York, 1956), 3rd ed.
  6. E. A. Guillemin, Mathematics of Circuit Analysis (John Wiley & Sons, Inc., New York, 1951).
  7. C. Lanczos, Linear Differential Operators (D. van Nostrand Inc., Princeton, New Jersey, 1962).
  8. C. Lanczos, Applied Analysis (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1956).

1962 (1)

1896 (1)

Rayleigh (J. W. Strutt), Phil. Mag. 42, 167 (1896).

Rayleigh (J. W. Strutt), Phil. Mag. 42, 167 (1896).

1858 (1)

L. Foucault, Ann. l’Observ. Imp. Paris 5, 197 (1858).

Barakat, R.

Carslaw, H. S.

H. S. Carslaw, Introduction to the Theory of Fourier’s Series and Integrals (Dover Publications, Inc., New York, 1956), 3rd ed.

Foucault, L.

L. Foucault, Ann. l’Observ. Imp. Paris 5, 197 (1858).

Françon, M.

M. Françon, article in Handbuch der Physik, edited by S. Flügge (Springer-Verlag, Berlin, 1956), Vol. XXIV, pp. 171, 342; A. Marechal and M. Françon, Diffraction—Structure des images (Editions de la Revue d’Optique, Paris, 1960), pp. 38, 50, 169.
[Crossref]

Guillemin, E. A.

E. A. Guillemin, Mathematics of Circuit Analysis (John Wiley & Sons, Inc., New York, 1951).

Lanczos, C.

C. Lanczos, Linear Differential Operators (D. van Nostrand Inc., Princeton, New Jersey, 1962).

C. Lanczos, Applied Analysis (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1956).

Morello, M. V.

Rayleigh,

Rayleigh (J. W. Strutt), Phil. Mag. 42, 167 (1896).

Strutt, J. W.

Rayleigh (J. W. Strutt), Phil. Mag. 42, 167 (1896).

Ann. l’Observ. Imp. Paris (1)

L. Foucault, Ann. l’Observ. Imp. Paris 5, 197 (1858).

J. Opt. Soc. Am. (1)

Phil. Mag. (1)

Rayleigh (J. W. Strutt), Phil. Mag. 42, 167 (1896).

Other (5)

M. Françon, article in Handbuch der Physik, edited by S. Flügge (Springer-Verlag, Berlin, 1956), Vol. XXIV, pp. 171, 342; A. Marechal and M. Françon, Diffraction—Structure des images (Editions de la Revue d’Optique, Paris, 1960), pp. 38, 50, 169.
[Crossref]

H. S. Carslaw, Introduction to the Theory of Fourier’s Series and Integrals (Dover Publications, Inc., New York, 1956), 3rd ed.

E. A. Guillemin, Mathematics of Circuit Analysis (John Wiley & Sons, Inc., New York, 1951).

C. Lanczos, Linear Differential Operators (D. van Nostrand Inc., Princeton, New Jersey, 1962).

C. Lanczos, Applied Analysis (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1956).

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Figures (16)

Fig. 1
Fig. 1

Square-wave and sine-wave objects of the same spatial frequency.

Fig. 2
Fig. 2

Fourier series representation of square wave illustrating the Gibbs phenomena.

Fig. 3
Fig. 3

Fourier series representation of square wave employing the Lanczos sigma factors.

Fig. 4
Fig. 4

Distribution of illuminance in square-wave image for a circular aperture suffering a quarter wave of defocusing.

Fig. 5
Fig. 5

Modulation of square and sine waves for an aberration-free slit aperture in focus.

Fig. 6
Fig. 6

Modulation of square and sine waves for an aberration-free circular aperture in focus.

Fig. 7
Fig. 7

Modulation of square and sine waves for an aberration-free slit aperture defocused W2=0.25 wavelength.

Fig. 8
Fig. 8

Modulation of square and sine waves for an aberration-free circular aperture defocused W2=0.25 wavelength.

Fig. 9
Fig. 9

Modulation of square and sine waves for an aberration-free slit aperture defocused W2=0.50 wavelength.

Fig. 10
Fig. 10

Modulation of square and sine waves for an aberration-free circular aperture defocused W2=0.50 wavelength.

Fig. 11
Fig. 11

Modulation of square and sine waves for an aberration-free slit aperture defocused W2=0.75 wavelength.

Fig. 12
Fig. 12

Modulation of square and sine waves for an aberration-free circular aperture defocused W2=0.75 wavelength.

Fig. 13
Fig. 13

Modulation of square and sine waves for an aberration-free slit aperture defocused W2=1.0 wavelength.

Fig. 14
Fig. 14

Modulation of square and sine waves for an aberration-free circular aperture defocused W2=1.0 wavelength.

Fig. 15
Fig. 15

Images of square-wave object for a circular aperture defocused W2=0.75 wavelength. Note the spurious resolution in last two frames.

Fig. 16
Fig. 16

Modulation of square and sine waves for a Tessar lens as determined theoretically from lens design data.

Tables (1)

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Table I Number of harmonic components of square wave which are allowed by optical system as a function of the spatial frequency range.

Equations (8)

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I ( ω ) = T ( ω ) O ( ω ) ,
O ( x ) = 1 2 + 2 π n = 0 ( - 1 ) n ( 2 n + 1 ) cos [ ( 2 n + 1 ) π L x ] ,
O ¯ N ( x ) = 2 N 2 L x - L / 2 N x + L / 2 N O N ( x ) d x ,
O N ( x ) = 1 2 + 2 π n = 0 ( - 1 ) n ( 2 n + 1 ) cos [ ( 2 n + 1 ) π L x ] .
O ¯ N ( x ) = 1 2 + 2 π n = 0 N ( - 1 ) n σ 2 n + 1 ( N ) ( 2 n + 1 ) × cos [ ( 2 n + 1 ) π L x ] ,
σ 2 n + 1 ( N ) = sin [ ( 2 n + 1 ) π / 2 N ] [ ( 2 n + 1 ) π / 2 N ] .
I ( x ) = 1 2 T ( O ) + 2 π n = 0 N ( - 1 ) n ( 2 n + 1 ) σ 2 n + 1 ( N ) × T [ ( 2 n + 1 ) ω 0 ] cos [ ( 2 n + 1 ) ω 0 x ] ,
modulation = ( I max - I min ) / ( I max + I min ) ,